Boundary Control Problems For Two Classes Of PDE-ODE Cascaded System

In this paper,the control problems of the heat-ODE system cascaded at two intermediate points and the Kd V-ODE cascaded system with Neumann boundary are studied.By using the Backstepping method,the feedback control laws are de-signed,and the exponential stabilization and null controllability of the corresponding closed-loop systems are obtained,respectively.In the first chapter,we mainly introduce the research background and research status,as well as the basic definitions and conclusions used in the subsequent proofs.In chapter 2,we study the exponential stabilization of a heat-ODE system cascaded at two intermediate points.Firstly,the Backstepping method is used to design a stable feedback control law,and the existence and~2smooth of the kernel function in the forward and inverse transformations are proved.Then,the expo-nential stabilization of the closed-loop system is proved,based on the stabilization of the target system and the invertibility of Backstepping transformation.Finally,the compatibility between the assumption in the literature and the controllability assumption in this paper is verified for the special case of=0 in the original system.In chapter 3,we study the null controllability of Kd V-ODE cascaded system with Neumann boundary.Firstly,the time is divided into time series intervals,and a Backstepping transformation and the corresponding target system are constructed on each time interval to obtain the norm estimates of the kernel function sequence and the feedback control law sequence.Secondly,the uniform decay rate estimation of the target system is obtained by constructing a suitable Lyapunov function on each time interval.Finally,the uniform norm estimation of the control law sequence of the Kd V-ODE cascaded system is obtained,and the null controllability of the Kd V-ODE cascaded system is proved by using the invertibility of Backstepping transformation.